Abstract

We report analytic formulas for the elements of the 2×2 cross-spectral density matrix of a stochastic electromagnetic anisotropic beam propagating through the turbulent atmosphere with the help of vector integration. From these formulas the changes in the spectral density (spectrum), in the spectral degree of polarization, and in the spectral degree of coherence of such a beam on propagation are determined. As an example, these quantities are calculated for a so-called anisotropic electromagnetic Gaussian Schell-model beam propagating in the isotropic and homogeneous atmosphere. In particular, it is shown numerically that for a beam of this class, unlike for an isotropic electromagnetic Gaussian Schell-model beam, its spectral degree of polarization does not return to its value in the source plane after propagating at sufficiently large distances in the atmosphere. It is also shown that the spectral degree of coherence of such a beam tends to zero with increasing distance of propagation through the turbulent atmosphere, in agreement with results previously reported for isotropic beams.

© 2007 Optical Society of America

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  1. M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
    [CrossRef]
  2. S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1304 (1979).
    [CrossRef]
  3. Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source beam," Radio Sci. 18, 551-556 (1983).
    [CrossRef]
  4. J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
    [CrossRef]
  5. J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
    [CrossRef]
  6. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  7. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  8. T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
    [CrossRef]
  9. G. P. Berman and A. A. Chumak, "Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere," Phys. Rev. A 74, 013805 (2006).
    [CrossRef]
  10. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  11. E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett. 28, 1078-1080 (2003).
    [CrossRef] [PubMed]
  12. H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
    [CrossRef]
  13. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
    [CrossRef]
  14. O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
    [CrossRef]
  15. X. Ji, E. Zhang, and B. Lü, "Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere," Opt. Commun. 275, 292-300 (2007).
    [CrossRef]
  16. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves in Random Media 14, 513-523 (2004).
    [CrossRef]
  17. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves in Random and Complex Media 15, 353-364 (2005).
    [CrossRef]
  18. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007).
    [CrossRef]
  19. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
    [CrossRef]
  20. H. Wang, X. Wang, A. Zeng, and K. Yang, "Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation," Opt. Lett. 32, 2215-2217 (2007).
    [CrossRef] [PubMed]
  21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  22. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).
  23. H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399-1406 (1972).
    [CrossRef] [PubMed]
  24. Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
    [CrossRef]

2007 (4)

X. Ji, E. Zhang, and B. Lü, "Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere," Opt. Commun. 275, 292-300 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

H. Wang, X. Wang, A. Zeng, and K. Yang, "Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation," Opt. Lett. 32, 2215-2217 (2007).
[CrossRef] [PubMed]

2006 (2)

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

G. P. Berman and A. A. Chumak, "Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere," Phys. Rev. A 74, 013805 (2006).
[CrossRef]

2005 (2)

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves in Random and Complex Media 15, 353-364 (2005).
[CrossRef]

2004 (2)

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves in Random Media 14, 513-523 (2004).
[CrossRef]

2003 (4)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett. 28, 1078-1080 (2003).
[CrossRef] [PubMed]

H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

2002 (2)

1991 (1)

J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

1990 (1)

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

1983 (1)

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source beam," Radio Sci. 18, 551-556 (1983).
[CrossRef]

1979 (1)

1977 (1)

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

1972 (1)

Baykal, Y.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source beam," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Belenkii, M. S.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Berman, G. P.

G. P. Berman and A. A. Chumak, "Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere," Phys. Rev. A 74, 013805 (2006).
[CrossRef]

Boardman, A. D.

J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Cai, Y.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Chumak, A. A.

G. P. Berman and A. A. Chumak, "Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere," Phys. Rev. A 74, 013805 (2006).
[CrossRef]

Davidson, F. M.

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves in Random and Complex Media 15, 353-364 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves in Random Media 14, 513-523 (2004).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

Eyyuboglu, H. T.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

Gbur, G.

He, S.

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Ji, X.

X. Ji, E. Zhang, and B. Lü, "Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere," Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Kon, A. I.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Korotkova, O.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves in Random and Complex Media 15, 353-364 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves in Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Lü, B.

X. Ji, E. Zhang, and B. Lü, "Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere," Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Mironov, V. L.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Plonus, M. A.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source beam," Radio Sci. 18, 551-556 (1983).
[CrossRef]

S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1304 (1979).
[CrossRef]

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

Ricklin, J. C.

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves in Random and Complex Media 15, 353-364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves in Random Media 14, 513-523 (2004).
[CrossRef]

Shirai, T.

Wang, H.

Wang, S. C. H.

Wang, S. J.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source beam," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Wang, X.

Wolf, E.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves in Random and Complex Media 15, 353-364 (2005).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves in Random Media 14, 513-523 (2004).
[CrossRef]

H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett. 28, 1078-1080 (2003).
[CrossRef] [PubMed]

T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

Wu, J.

J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

Yang, K.

Yura, H. T.

Zeng, A.

Zhang, E.

X. Ji, E. Zhang, and B. Lü, "Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere," Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

J. Mod. Opt. (3)

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (3)

H. Roychowdhury and E. Wolf, "Determination of the electric cross-spectral density matrix of a random electromagnetic beam," Opt. Commun. 226, 57-60 (2003).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

X. Ji, E. Zhang, and B. Lü, "Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere," Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Phys. Rev. A (1)

G. P. Berman and A. A. Chumak, "Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere," Phys. Rev. A 74, 013805 (2006).
[CrossRef]

Radio Sci. (1)

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source beam," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Sov. J. Quantum Electron. (1)

M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulent distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Waves in Random and Complex Media (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves in Random and Complex Media 15, 353-364 (2005).
[CrossRef]

Waves in Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves in Random Media 14, 513-523 (2004).
[CrossRef]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).

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Figures (4)

Fig. 1.
Fig. 1.

Illustrating the notation relating to propagation of a stochastic electromagnetic beam through the turbulent atmosphere.

Fig. 2.
Fig. 2.

Changes in the normalized spectral density S/S0 along the z-axis of anisotropic electromagnetic beams through the turbulent atmosphere with different C2 n . The source is assumed to be electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, Ax =2, Ay =1, Bxy =0.2exp(/3), σx =1cm, σy =2cm, δxx =δyy =2mm, δxy =3mm. The unit of C2 n is m-2 3.

Fig. 3.
Fig. 3.

Changes in the spectral degree of coherence µ along the z-axis of anisotropic electromagnetic Gaussian Schell-model beams propagating through the turbulent atmosphere with different C2 n . The source parameters are the same as Fig. 2. Pairs of field points: (a) ρ T 12=(1mm, 0, -1mm, 0), (b) ρ T 12=(3 mm, 0, -3 mm, 0) The unit of C2 n is m-2/3.

Fig. 4.
Fig. 4.

Changes in the spectral degree of polarization P along the z-axis of anisotropic electromagnetic Gaussian Schell-model beams propagating through the turbulent atmosphere with different C2 n . The source parameters are the same as in Fig. 2, but (a) σx =σy 1 cm, (b) σx =1 cm, σy =2 cm. The unit of C2 n is m-2/3.

Equations (19)

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W ( r 1 , r 2 , ω ) [ W ij ( r 1 , r 2 , ω ) ] = [ E i * ( r 1 , ω ) E j ( r 2 , ω ) ] , ( i = x , y ; j = x , y ) .
E i ( ρ , z , ω ) = ik 2 π z exp ( ikz ) E i ( 0 ) ( ρ , ω ) exp [ ik 2 z ( ρ ρ ) 2 ] exp [ ψ ( ρ , ρ , z , ω ) ] d 2 ρ ,
W ij ( ρ 1 , ρ 2 , z , ω ) = k 2 4 π 2 z 2 W ij ( 0 ) ( ρ 1 , ρ 2 , ω ) exp [ ik 2 z ( ρ 1 ρ 1 ) 2 + ik 2 z ( ρ 2 ρ 2 ) 2 ]
× exp [ ψ * ( ρ 1 , ρ 1 , z , ω ) + ψ ( ρ 2 , ρ 2 , z , ω ) ] m d 2 ρ 1 d 2 ρ 2 ,
exp [ ψ * ( ρ 1 , ρ 1 , z , ω ) + ψ ( ρ 2 , ρ 2 , z , ω ) ] m = exp [ ( 1 2 ) D ψ ( ρ d , ρ d ) ]
exp [ ( 1 ρ 0 2 ) ( ρ d 2 + ρ d · ρ d + ρ d 2 ) ] ,
W ij ( ρ 12 , z , ω ) = k 2 4 π 2 [ Det ( B ¯ ) ] 1 2 W ij ( 0 ) ( ρ 12 , ω )
× exp [ ik 2 ( ρ 12 T B ¯ 1 ρ 12 2 ρ 12 T B ¯ 1 ρ 12 + ρ 12 T B ¯ 1 ρ 12 ) ] ,
× exp [ ik 2 ( ρ 12 T P ¯ ρ 12 + ρ 12 T P ¯ ρ 12 + ρ 12 T P ¯ ρ 12 ) ] d 4 ρ 12
B ¯ = [ z I 0 0 z I ] , P ¯ = 2 ik ρ 0 2 [ I I I I ] ,
W ij ( 0 ) ( ρ 1 , ρ 2 , ω ) = S i ( 0 ) ( ρ 1 , ω ) S j ( 0 ) ( ρ 2 , ω ) η ij ( 0 ) ( ρ 2 ρ 1 , ω ) ,
W ij ( 0 ) ( ρ 1 , ρ 2 , ω ) = A i A j B ij exp ( ρ 1 2 4 σ i 2 ρ 2 2 4 σ j 2 ) exp ( ρ 2 ρ 1 2 2 δ ij 2 ) .
W ij ( 0 ) ( ρ 12 , ω ) = A i A j B ij exp ( ik 2 ρ 12 T M ij 1 ρ 12 ) ,
M ij 1 = [ i 2 k σ i 2 i k δ ij 2 0 i k δ ij 2 0 0 i 2 k σ i 2 i k δ ij 2 0 i k δ ij 2 i k δ ij 2 0 i 2 k σ j 2 i k δ ij 2 0 0 i k δ ij 2 0 i 2 k σ j 2 i k δ ij 2 ]
W ij ( ρ 12 , z , ω ) = A i A j B ij [ Det ( I ¯ + BP ¯ + B ¯ M ij 1 ) ] 1 2 exp { ik 2 ρ 12 T [ ( B ¯ 1 + P ¯ )
( B ¯ 1 1 2 P ¯ ) T ( B ¯ 1 + P ¯ + M ij 1 ) 1 ( B ¯ 1 1 2 P ¯ ) ] ρ 12 }
S ( ρ , z , ω ) = Tr W ( ρ , z , ω ) ,
μ ( ρ 12 , z , ω ) = Tr W ( ρ 12 , z , ω ) Tr W ( ρ 11 , z , ω ) Tr W ( ρ 22 , z , ω ) ,
P ( ρ , z , ω ) = 1 4 Det W ( ρ , z , ω ) [ Tr W ( ρ , z , ω ) ] 2 .

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